I would like to recast Nik Weaver's question and Harvey
Friedman's response to it,
in order to get a better understanding of the differences between
them regarding
predicativity.
Nik Weaver wrote:
> Let us grant [that for] the predicativist...for each a he has some
> way to
> make the deduction
>> (*) from I(a) and Prov_{S_a}(A(n)), infer A(n)
>> for any formula A, where I(a) formalizes the assertion that a is an
> ordinal notation (supporting transfinite induction for sets).
>> Shouldn't he then accept the assertion
>> (**) (forall a)(forall n)[I(a) and Prov_{S_a}(A_n) --> A(n)]
>> for any formula A?
>
> If my critique is truly fallacious, surely someone can explain
>> (1) why (*) is reasonable
>> (2) why (**) is not reasonable.
>
So, on the basis of the above, I would expect those who object
to Weaver's analysis to simply answer why (*) is reasonable
(predicatively) and (**) is not reasonable.
In response to Nik Weaver's question,
Harvey Friedman wrote:
*****
"7. Weaver MUST be claiming that *) implies **), in some relevant
sense of
implies.
for how else can he claim to have "refuted" Feferman/Schutte?
I pointed out that not only does *) not imply **) mathematically, but
also
*) does not imply **) philosophically either. It would be both a
mathematical and philosophical mistake. I see no relevant sense of
implies
here. This situation is completely familiar and very common throughout
f.o.m."
********
Earlier, Nik Weaver wrote, if memory serves, that he is *not*
contending
that (*) implies (**). So, in this sense, I take Nik Weaver to be
agreeing with
Harvey Friedman. So I see Weaver as asking whether it is
(predicatively) "reasonable" to accept (*) but not to accept (**)--
*knowing*
that no implication exists from (*) to (**).
I am writing this simply for clarification and to achieve a better
understanding of predicativity. So, I am requesting that persons
who have a "reasonably strong sense" of what is predicative and
what is not to please offer their opinions about the
"reasonability question" concerning (*) and (**)? Since I
have only very vague ideas (from courses taken long ago)
about such things as "vicious circles" and "harmless circles", I
myself am in no position to offer an opinion.
Thank you.
Charlie Silver